Abstract

Over the last decades a great deal of attention has been given to the existence of invariant probability measures for Markov chains. In general it is not a simple matter to determine whether a given Markov chain on a general state space has an invariant probability measure. For discrete-time Markov chains there exist sufficient conditions based on the Foster-Lyapunov type criterion. This Foster-Lyapunov criterion, also known as the drift condition, is written in terms of a set C, called the test set. Besides the Foster-Lyapunov criterion being satisfied, these sufficient conditions require some assumption on the test set C and other additional hypotheses. In the first part of this talk, it will be shown that the Foster-Lyapunov criterion is sufficient to ensure the existence of an invariant probability measure without any additional hypotheses (such as irreducibility). In the second part of this talk, we shall study the question as to whether the Foster-Lyapunov criterion is a necessary condition. This will motivate the presentation of a new assumption which generalizes the concept of a T-chain.
 

Convenor:Aidan Sudbury